The 24-year-old is 15th in the drivers' standings, having achieved his first podium with Williams at the Belgian Grand Prix, while also recording top-10 finishes in Hungary, Italy and Russia.
Many expect Russell to offer a genuine challenge to Hamilton, who recently signed a new two-year deal with Mercedes, with the latter seeking an eighth world title and just eight points behind leader Max Verstappen heading into the final two races of the 2021 season.
The 36-year-old is anticipating a healthy rivalry with his compatriot and hopes he can be a positive influence on the 2018 Formula 2® world champion.
"You've seen George is hugely respectful," he said in an interview published by BBC Sport.
"He's a super-talented young man and I think there's a huge amount of respect already going in, and we've got a nice balance at the moment.
"But he's going to want to be quick, he's going to want to show up and win, and do all those things that you do when you enter a new role.
"I remember in 2007 when I went up against [Fernando] Alonso [at McLaren]. Of course, I wanted to beat him at the first race, so I appreciate and expect George to have that mentality; otherwise, he's not a winner.
"But I'm in a different place. I really want to see him succeed. There's going to be a point where I don't continue in this sport. He's my teammate, and he's going to be the next Brit that I want to see win a world championship.
"So, while we are going to be competing, and I want to win on track. I really hope I can have a positive influence on how he conducts himself within the team, whether it's the time he commits to engineering or how he churns through the data, or even just how he drives on track."